Chapter 12: Functional Analysis, Hilbert Spaces and Wavelets

Abstract:

A Hilbert space is a set, ℋ of elements, or vectors, (f, g, h, …) which satisfies the following conditions (1) to (5).

(1) If f and g belong to ℋ, then there is a unique element of ℋ, denoted by f + g, the operation of addition (+) being invertible, commutative and associative.

(2) If c is a complex number, then for any f in ℋ, there is an element cf of ℋ; and the multiplication of vectors by complex numbers thereby defined satisfies the distributive conditions

(3) Hilbert spaces ℋ possess a zero element, 0, characterized by the property that 0 + f = f for all vectors f in ℋ.

(4) For each pair of vectors f, g in ℋ, there is a complex number 〈f|g〉, termed the inner product or scalar product of f with g, such that

and Equality in the last formula occurs only if f = 0. The scalar product defines the norm ‖f‖ = 〈f|f〉^1/2.

(5) If {f_n} is a sequence in ℋ satisfying the Cauchy condition that

as m and n tend independently to infinity, then there is a unique element f of ℋ such that ‖f_n − f‖ → 0 as n → ∞…